cheesemonkey wonders

cheesemonkey wonders

Monday, November 12, 2018

Proof Portfolios: Revenge of the Immersive Project

Some years ago, in a town about 11 miles south of here, I taught both math and English (I'm credentialed in both — shhh... don't tell anybody). Our English department was the single most functional academic department I have ever been a part of. When you came right down to it, what we did wasn't rocket science. We had a method, we used the method, and we assessed together by grade level. The 8th grade team consisted of Alec MacKenzie, Kelly Starnes, Linda Grady, and me. We are all still in the classroom, which strikes me as a minor miracle.

The writing method we used was a combination of the Jane Shaffer method and Six Traits. Every teacher in the district had been trained on the method, and collaborative rubrics and projects had been developed over the years. The method was flexible, but we all agreed on certain basic components. We used Jane Shaffer's method of color-coding, from elementary through middle school. This meant that every student in the district developed a common understanding of what a topic sentence is, what a claim is, how we use evidence to support our claims, and how we use reasoning to tie things together. I believe it is still one of the highest-performing writing districts in California.

One of our signature practices was that we gave a fall writing assessment and a spring writing assessment. In middle school it was a two-day affair, tied to the literature curriculum. Time was allocated for pre-writing and writing.

And then we teachers were given an on-campus release day so we could read and score holistically — together. We double-scored each sample, using a rubric and highlighters.

It drove us crazy, but it also enabled us to see patterns. And because we could see patterns, we could adapt instruction to address the gaps or needs we identified.

Shouldn't this be the norm for instruction?

I have long wanted to use this approach with my teaching of proof in high school, but this was the first year I got my act together to run my own personal pilot program. I don't have a colleague with whom to work on this, so I went it alone. I created a four-day project that and gave them one day's worth of stuff to work on each day.
Here is the zip file with all four days' worth of assignments:
The secret of doing an assignment like this is radical: you have to relinquish control. You cannot be the only one giving students feedback. In fact, there is so much practice here, it is completely impossible. That is good. One thing I have learned as a writer and as a teacher of writing is that you learn how to write by writing a lot. The same is true with proof and proving. Students need space to immerse themselves and not worry about whether every mark they make is "right" or "wrong."

So each day had its "stuff." Four small proofs a day, plus reflection and peer review. Then more the next day.

The complaints and lamentations were filled with drama. "OH MY GOD, DR. S — THAT ASSIGNMENT WAS HARD." But they could tell that they had accomplished something.

My assessment strategy was to be rigorous about completion but merciful with points. It was only worth a quiz grade (100 points), and my default score for students who completed every section was a 95. There are rewards for following instructions. Missing sections or components left blank cost more points.

But none of that matters. My goal was to get students doing a LOT of proof -- writing shitty first drafts, comparing notes with each other, and using a rubric to assess each other's work. Dogen Zenji said, "When you walk in the mist, you get wet."

And it seems to have made a difference.

I am excited to see what happens on the next major test that includes a proof. Photos of student work to follow.

As always, let me know what you think and what happens if you use this!

Tuesday, November 6, 2018

Proof Portfolios

Over the last five years of teaching proofs in Geometry, I have learned two things: (1) the most effective student understanding comes from writing about their proof process, not from the proving itself, and (2) the most effective feedback process for students is a peer-to-peer reciprocal feedback process.

So this year, when I had to be out of school for a few days, I designed a Proof Portfolio project for them to do in my absence.

Each day had four small, reasonable proofs students had to do — and they could collaborate on these. But then... they had to write a number of short-answer reflections to analysis questions based on their own proofs in the day's set.

In addition, they had to find a peer to trade with and to give a rubric-based peer review and reflection.

In my class, they did this for several consecutive days. I made it worth a quiz/project grade.

When I returned, there was a great deal of wailing and moaning and gnashing of teeth about How Hard This Project Was and How Hard They All Worked.

It was clear that this project was a rite of passage for my classes.

But as I'm reading their work, I am blown away by how much they seem to have learned!

Their mastery of proof is not perfect. But it is authentic and it is growing. And to me, that is the most important point at this stage.


I made up four days' worth of activities. Each day is two double-sided pages (proofs & reflections).

Here is a link to the G-drive folder with the four PDFs:

Monday, September 3, 2018

The New Normal

By U.S. standards, San Francisco is a large, densely populated city, but geographically, we are really quite small — only 7 x 7, as the saying goes. We are also both a city and a county, the only one in California. Which means that everybody knows somebody at any given school.

Last week, a gun went off at in a classroom at one of our sister high schools across the city. This happened around 11:15 a.m. Naturally, the ninth-grade students in my classroom knew about it by around 11:20.
"It was a freshman." 
"It was in the bathroom." 
"No, it was in a classroom. They're on lockdown."
"He had a Spiderman backpack."
"My cousin goes to school there there."
 It was scary to know about while it was happening. Everybody does lockdown drills and everybody complains about them, but nobody would have objected if it had been a lockdown for real in our own school.

After a few minutes of frantic following the thread of what was known, I turned off the lights. "Let's stop and take a moment to meditate and send them good energy."

One boy with a blue sweatshirt and a worried look said, "Like, should we pray for them?"

I said, "If that is what your heart tells you to do in this moment, then you should do that." He nodded, closed his eyes, and laced his fingers together with practiced intensity.

I guided students into a brief mindfulness meditation. We laid our phones face down on our desks, and I gave them the instruction on how to do meditation, focusing their attention on their breath coming in and out at their nose. Thirty-six wired, anxious fourteen-year-olds and I spent the next two minutes anchoring our crazy, overstimulated monkey minds together in our breath.

I felt the mood loosen in the room; then I flipped the light back on and we returned to our lesson.

I kept an eye on my Twitter feed, and when class was drawing to a close, I told them that the police had apparently secured the gun and the area and that they were just waiting for the all clear.

A few years ago, I would have lost the focus of the students and not been able to redirect it. Now I know better what to do and how to do it.

I just wish this weren't happening as much as it seems to be in our country.

Saturday, August 25, 2018

First week and AVID strategies

I made it through the first week! And the chairs never had to get unstuck from the floor!

I did so much more metacognitive work this week than I’ve ever done before. Every day we did a say-do-mean summary at the end (My notes say... This enables me to... This means that...). Every day I tied our essential question to our work and to our metacognitive goals. And every day I used 10-2 processing to keep the pace up and get kids collaborating instead of relying on me. For every ten minutes of notes, I gave two minutes of processing time to catch up and collaborate on making their notes accurate. When my Geometry students asked what the terms were going to be on Friday’s vocabulary quiz, I didn’t answer that question. Instead, I instructed them to take two minutes to compare notes at their tables. See if anybody caught something you missed. Make sure everybody has everything they should have in their notes.

And they did it.

This was a powerful learning for me.

Until now, I used to answer those questions.

Now I am encouraging self-reliance and resourcefulness and a thinking classroom instead. I am doing this every place I can.

It’s a small instructional shift towards resourcefulness, but it feels seismic. I definitely want persistence, but not thoughtless persistance. I want to cultivate thoughtful persistence and resourceful persistance. You have super-smart classmates. Use them as an additional resource. Use them as a primary resource.

I felt so proud of my kids this week, it gave me energy. Even though I was exhausted, it gave me energy.

Now I am sitting in my favorite place in the world with my sofa and my fireplace and my music and my dog. The sun is burning away at the fog. The beginnings of a good new year.

Sunday, August 12, 2018

A Course in Thinking

This blog post is also a session at Sam Shah's The Virtual Conference of Mathematical Flavors 

In a conference on flavors of mathematical teaching and learning, you could be forgiven for expecting every session to address some version of the age-old arguments about whether there are math people and non-math people, about whether it is better to have a growth mindset or a fixed mindset, or about whether mathematics is the most beautiful of all the disciplines we teach in school.

Which is probably why I feel so hesitant to confess my dirty little secret. In my classroom, I teach mathematics as one of the humanities.

The fact is that math is a human activity. If you are human, you cannot escape it. And what I have come to value the most about the opportunity to teach mathematics is that it has become one of the most pivotal ways in which we transmit the culture and values we cherish the most. For me, some of those values include respect, communication, empathy, understanding, persuasion, civil disagreement, persistence, deep listening, reassessing, and changing one’s mind. Math class gives me an opportunity to share all of these aspects of being human and living together in human culture.

So I’ve rewritten the introduction to my course syllabus to emphasize some of the things about which I feel most strongly — and which I believe are the most powerful and important things I have to share with my students over the course of the next year.

I listen to hear your thoughts.

This is a course about thinking.

You are here to learn how to think better and to use your thinking to accomplish things in the world.

The essence of thinking is sense-making. To make sense of things, you have to understand them, which means you have to want to understand them. One of our mottos in this class is, You gotta wanna.  This is as important in mathematics as in everything else.

So everything in this class is about making sense of things. In mathematics — as in life — we mostly make sense of problems. If you do not yet know that life presents a steady stream of problems to be solved, you will soon. 

In this class, we happen to use mathematics  as a ground for thinking. I will tell you a secret up front: I don’t actually care if you  ever “use” this stuff ever again or not... so please don’t waste time asking me that question. It is a boring and senseless question. What I care about passionately is that you learn how to think and communicate at a more advanced level than you are capable of right now. 

And that is what we are going to work on.

Thinking better is a set of skills you can actually learn and use at this school. It is the appropriate focus of math class.

Since you are going to be thinking for the rest of your lives, you are going to need to make sense of things you don’t initially understand. And then you’re going to have to persuade other people that your thinking is right. So your goal in this course should be to grow as an active sense-maker who is skilled in using these tools of thinking.

You should also learn to treat your thinking with respect. The mind is a muscle, and this school is a place where we work to strengthen our thinking muscles. That means we need to develop strength, flexibility, and endurance in our thinking — in other words, you need to become a strong thinker, a flexible thinker, and a persistent thinker. You also need to become a good collaborator, which means you need to become a better listener.

While there are no guarantees, I can promise you that if you focus on these goals here, you will do well in this class, and these skills will carry you very far in your life.

Sunday, July 29, 2018

TMC 18 Recap: a lesson from the Rock and Roll Hall of Fame: Find what you need. Refuse to be stopped.

On my last day in Cleveland, I went to the Rock and Roll Hall of Fame. This was my one assignment from my husband David, a long-time jazz, rock, and world music radio DJ who has not yet had his own reason to visit Cleveland.

At the end of the third floor Hall of Fame exhibit is the "Power of Rock Experience," a small, stadium-style theater with a big-screen, state-of-the-art showing of Jonathan Demme's highlight film of Hall of Fame Induction concerts. Strobe, lighting effects, and fog machines give you a powerful, close-up concert experience, even though you are in a small, stadium-style theater in Cleveland.

What stayed with me was the climax of the film, the joyful, posthumous concert tribute to George Harrison, in which Tom Petty, Prince, George's son Dhani, and a number of other amazingly famous RRHOF inductees gave their own burning-down-the-house tribute version of While My Guitar Gently Weeps. It was a killer tribute to a killer song that has meant a lot to me in my own life. Prince's guitar solo, tearing up Eric Clapton's original version on the Beatles' recording, blew my mind.

When I called my husband later that night to tell him about it, David said, "Isn't that the one where Prince throws his guitar into the air and it never comes down?"

I laughed and said, "Yep, that was it."

That's when it hit me—the lesson I have taken away from my seven years of attending Twitter Math Camp: Find what you need. Refuse to be stopped.

The story of that song, as I understand it, was and remains amazing to me. George was frustrated by his inability to get his songs onto the Beatles' albums. He felt like he couldn't get John and Paul to pay attention to this song and he vented to everybody he knew. Finally, after venting to his good friend Eric Clapton, Eric came to their next recording session to support George. He sat in with them, played the guitar solos that made them really hear the power of the song, and the rest was history.

Seven years ago, when we conceived and held the first Twitter Math Camp in St. Louis, nobody noticed us. Nobody cared. We were 39 North American math teachers who had a yearning for community. A couple people came from Canada. One person came from Amman, Jordan. I was the only person who came from California. In fact, we had more participants from Mississippi than we had from California.

And we created something powerful, something we needed.

It was a stone soup effort, and it still is. What people may not know is that this is something that was created out of thin air. A core group of about 15 of us agreed to provide the core structure. There was no organization, no staff at that time, except for Jason Henry, who was about to join an effort he didn't even know much about.

People just showed up and brought what they had.

And what they had to share was amazing.
I learned a lot from that experience, but the most important thing I learned was what I consider to be the essence of Twitter Math Camp:
Complain and vent if you must, but then find what you need and refuse to be stopped.
Know that there is going to be loneliness, there is going to be heartache, and there is going to be risk.

But also know that there is going to be a miracle.

When the Beatles started, they were four dudes from Liverpool who wanted to play rock and roll so much that they spent their adolescences doing what they loved, making music and playing for people in dark, dank basement clubs in Germany.

They did this over and over and over — for years.

They had no idea that what they were going to become was THE BEATLES. They had found what they loved, and they committed themselves to honing their craft. They refused to be stopped.

By the time they got their first recording contract, they had gotten so good they could bang an amazing album out in just a few takes.

They kept doing it even after it became tougher to do.

Eventually, they gave up giving live concerts altogether. The thing they had loved the most had become dangerous and damn near impossible. But they never stopped loving it. In the last "public" concert they gave together, on the roof of Apple Records in London, they played to the sky and to anybody within earshot who could hear them. Fortunately for us, one of their managers had the foresight to film this event, which occurred not long before they broke up forever.

But for me, sitting in the darkened theater, watching Tom Petty and Prince and Dhani Harrison bring down the house with this performance of a song that almost never saw the light of day, there was joy in remembering how this same spirit of determination brought me this annual retreat/conference/event that has become so dear to me.

Find what you need. Refuse to be stopped.

Here are some of the other important secrets I have learned from seven years of TMC.

Get busy and recognize belonging as blessing. Find what you need and need to share. Refine your own craft. Trust that people like you need you and are searching for you. If one group is unable to see and value what you need yet, refuse to be stopped. Continue under all circumstances and keep searching for your people. If you haven't found them yet, keep searching.

We were nobody. I'm still not sure how we became somebody. I remember The Great Facebook Friending of Winter Break 2011, when I was certain that I was going to wake up and discover why you should never friend and meet people you have only ever met on the internet. I was sure I was going to have to go into Facebook Witness Protection. But it turned out OK. In fact, I made a number of lifelong friends that way.

So keep following and keep friending. No risk, no reward. Remember that the people you are looking for are also looking for you.

Practice Gratitude. Gratitude is a giant, holy yes that I keep saying over and over and over. When I found someone who was generous with their blog and their tweeted advice and their encouragement, I said thank you. My way of saying thank you in those earliest days was to say "thank you" over and over but when somebody was inconceivably generous with me, I knitted them a small, stellated dodecahedron and mailed it to them. I didn't ask. I didn't promise. I just did it. I did this thirteen times.

I've been just as blessed — and surprised — to receive things for my own generosity. During a dark time, Tina Cardone crocheted me a unicorn that now sits on my desk. @veganmathbeagle crocheted me an otter. @caseymcteach mailed me a monkey lanyard. Kristin Fouss gave me a Fiona the baby hippo t-shirt. And other treasures too numerous to elaborate here.

The point is, the practice of gratitude is a big part of the essence of Twitter Math Camp. It's one of the invisible threads that bind us together.

If you need this kind of connection, keep searching and keep practicing gratitude. It will come back to you many times over if you let it.

Be unconditionally constructive. This is especially important on Twitter itself. With all the negativity in our world right now, I think the best thing we can each do is to contribute unconditional positivity wherever we can. That's a big part of being a teacher, to be sure, but it's also a critical part of being a digital citizen.I am constantly trying to remember to ask myself, What am I contributing right now?

The Power of 'Yet.' Refuse to be discouraged. If people haven't come around to your point of view yet, recognize that you just haven't reached them or persuaded them yet. Keep going. Keep growing, Keep working. If you're right, you'll convince people eventually.

Make what you need. While #tmcjealousycamp is fun and funny, it's beside the point. What is important is to figure out what you need and push forward and make it. That is the power of un-conferences and salons and all the other forms of community. If you are committed to finding your community of math teachers, that's it. Don't let anybody stop you. Figure out what you need and create it. You'll be astonished at the support you get back from the Universe.

Friday, April 6, 2018

HERESY WARNING: Breaking through on quadratic function analysis & graph sketching

Over the last two days, I've had a few important breakthroughs with discouraged Algebra 1 learners about quadratic functions and their graphs. I wanted to document this for myself before I have a chance to forget about it for next year.

If you have rigid beliefs about the only ways for students to approach quadratic functions, analysis, and graphs, then this post is definitely not for you.

Consider yourselves warned.

My most discouraged Algebra 1 learners are extremely gifted kids, but this year's crop are definitely dreamers, not stalkers. They need to really marinate in something for a long time before it takes root in their minds. They are factoring warriors, but the quadratic formula and complicated answers can be really daunting for them.

The connection I have wanted all of my students to make between quadratic functions and their graphs is this: even when a quadratic doesn't have neat, simple, integer answers, it still has a number of neat, simple aspects that they can grab hold of. There will always be an axis of symmetry. There will always be a vertex.

What I have discovered — or rather, what they have been teaching me all this week — is that if you approach a quadratic function from the understanding that you already know how to find the AOS and the vertex, you can make a TON of important discoveries and understandings about its graph and the many properties of the function that are important to understand.

What they taught me today is that they know how to use the axis of symmetry formula like a key in a lock. They can identify a, b, and c in a quadratic, and they understand that they can find and graph the AOS equation quickly and fearlessly. Then they can plug in the value they found for the axis of symmetry to identify the vertex of the parabola.

From there, it is a simple matter to find some more points for your sketch and to identify their mirror reflections across the axis of symmetry.

The beauty of their method is that it makes it easy for them to develop a meaningful conjecture about whether or not a quadratic function has any zeros.

If the parabola is floating above the x-axis, then they can use the QF to confirm their hunch that there are no real-number zeros to the function. Likewise if the parabola is submerged below the x-axis.

I love that they naturally figured out today what it means for the vertex to be a maximum or a minimum.

And I especially love the fact that they made these discoveries themselves.

They still don't understand how to complete the square or use the quadratic formula to blast through problem after problem to find complicated zeros of non-trivial quadratics.

But this feels less important to me than the fact that they have made important connections and developed their own methods for investigating quadratic functions. And it has been an important reminder to me to design learning experiences that empower them to make these connections and discoveries for themselves.